Multifractal Turbulence in the Heliosphere and the...

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Multifractal Turbulence in the Heliosphere and the Heliosheath Wieslaw M. Macek Space Research Centre, Polish Academy of Sciences, Bartycka 18 A, 00-716 Warsaw, Poland; Faculty of Mathematics and Natural Sciences, Cardinal Stefan Wyszy´ nski University, oycickiego 1/3, 01-938 Warsaw, Poland; Dipartimento di Fisica, Universit ` a della Calabria, Rende-Cosenza, Italy. e-mail: [email protected], http://www.cbk.waw.pl/macek NWCW, La Jolla, 4-8 March 2013

Transcript of Multifractal Turbulence in the Heliosphere and the...

Page 1: Multifractal Turbulence in the Heliosphere and the Heliosheathusers.cbk.waw.pl/~macek/conf/recent/nwcw_2013.pdf · Carbone, 1993; Frisch, 1995). In particular, multifractal scaling

Multifractal Turbulence in the Heliosphereand the Heliosheath

Wiesław M. MacekSpace Research Centre, Polish Academy of Sciences,

Bartycka 18 A, 00-716 Warsaw, Poland;Faculty of Mathematics and Natural Sciences, Cardinal Stefan Wyszynski University,

Woycickiego 1/3, 01-938 Warsaw, Poland;Dipartimento di Fisica, Universita della Calabria, Rende-Cosenza, Italy.e-mail: [email protected], http://www.cbk.waw.pl/∼macek

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Plan of Presentation1. Introduction• Fractals and Multifractals• Solar Wind• Importance of Multifractality

2. Methods of Data Analysis• Magnetic Field Strength Fluctuations• Generalized Scaling Property• Measures and Multifractality• Multifractal Model for Magnetic Turbulence• Degree of Multifractality and Asymmetry

3. Results for Voyager spacecraft• Multifractal Singularity Spectrum• Degree of Multifractality and Asymmetry• Multifractal Turbulence at the Termination Shock

4. Conclusions

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Abstract

We present first results of the multifractal scaling of the fluctuations of the interplanetarymagnetic field strength as measured onboard Voyager 2 in the very distant heliosphereand even in the heliosheath. More specifically, we analyze the spectra observed byVoyager 2 in a wide range of heliospheric distances from 6 to 90 astronomical units (AU)that are compared with those of Voyager 1 already analyzed between 7 and 107 AU,which is now approaching the heliopause.

We focus on the singularity multifractal spectrum before and after crossing thetermination heliospheric shock by Voyager 1 at 94 AU and Voyager 2 at 84 AU fromthe Sun. It is worth noting that the spectrum is prevalently right-skewed inside the wholeheliosphere. Moreover, we have observed a change of the asymmetry of the spectrumat the termination shock.

We show that the degree of multifractality falls steadily with the distance from the Sun.In addition, the multifractal structure is apparently modulated by the solar activity, witha time shift of several years, corresponding to a distance of about 10 AU, resulting fromthe evolution of the whole heliosphere. Hence this basic result also brings significantadditional support to some earlier claims suggesting that the solar wind terminationshock is asymmetric.

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Prologue

A fractal is a rough or fragmentedgeometrical object that can be subdividedin parts, each of which is (at leastapproximately) a reduced-size copy ofthe whole. Fractals are generally self-similar and independent of scale (fractaldimension).

A multifractal is a set of intertwinedfractals. Self-similarity of multifractalsis scale dependent (spectrum ofdimensions). A deviation from astrict self-similarity is also calledintermittency.

Two-scale Cantor set.

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Importance of Multifractality

The general aim of the present paper is to report on the new developments in systems exhibitingturbulent behavior by using multifractals, with application to phenomenological approach tothis complex issue. Following the idea of Kolmogorov (1941) and Kraichnan (1965) variousmultifractal models of turbulence have been developed (Meneveau and Sreenivasan, 1987;Carbone, 1993; Frisch, 1995).

In particular, multifractal scaling of this flux in solar wind turbulence using Helios (plasma)data in the inner heliosphere has been analyzed by March et al. (1996). It is knownthat fluctuations of the solar magnetic fields may also exhibit multifractal scaling laws. Themultifractal spectrum has been investigated using magnetic field data measured in situ byVoyager in the outer heliosphere up to large distances from the Sun (Burlaga, 1991, 1995,2004) and even in the heliosheath (Burlaga and Ness, 2010; Burlaga et al., 2006).

To quantify scaling of solar wind turbulence we have developed a generalized two-scale weighted Cantor set model using the partition technique (Macek 2007; Macek andSzczepaniak, 2008), which leads to complementary information about the multifractal natureof the fluctuations as the rank-ordered multifractal analysis (cf. Lamy et al., 2010).

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We have investigated the spectrum of generalized dimensions and the correspondingmultifractal singularity spectrum depending on one probability measure parameter and tworescaling parameters. In this way we have looked at the inhomogeneous rate of the transferof the energy flux indicating multifractal and intermittent behavior of solar wind turbulence.In particular, we have studied in detail fluctuations of the velocity of the flow of the solarwind, as measured in the inner heliosphere by Helios (Macek and Szczepaniak, 2008),Advanced Composition Explorer (ACE) (Szczepaniak and Macek, 2008), and Voyager in theouter heliosphere (Macek and Wawrzaszek, 2009 ), including Ulysses observations at highheliospheric latitudes (Wawrzaszek and Macek, 2010).

Voyager 1 crossed the termination heliospheric shock, which separates the Solar Systemplasma from the surrounding heliosheath, with the subsonic solar wind, on 16 December2004 at heliocentric distances of 94 AU (at present its distance to the Sun is about 122 AUapproaching the heliopause). Please note that (using the pressure balance) the distance to thenose of the heliopause has been estimated to be ∼120 AU (Macek, 1998). Later, in 2007 alsoVoyager 2 crossed the termination shock at least five times at distances of 84 AU (now is at100 AU).

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Admittedly, variations of the magnetic field strength observed by Voyager 2 have also beenanalyzed including those prior and after crossing the termination shock up to distances of ∼90AU in 2009 (from 2007.7 to 2009.4), see (Burlaga et al., 2008, 2009, 2010). However, themultifractal spectrum using Voyager 2 data has only been analyzed at 25 AU by Burlaga (1991).but in a more distant heliosphere and especially near the termination shock this multifractalanalysis is still missing.

Therefore, the aim of our present study, which is a substantial extension of the previousGRL letter (2011) is to investigate the multifractal scaling for both Voyager 1 and 2 data that willallow us to infer new information about the multifractal structure of the heliospheric magneticfields in the northern and southern hemisphere, including the correlation with the solar cycle.

In particular, we show that multifractal structure is modulated by the solar activity withsome time delay and confirm that the degree of multifractality is also decreasing with distance:before shock crossing is greater than that in the heliosheath. Moreover, we demonstratethat the multifractal spectrum is asymmetric before shock crossing, in contrast to the nearlysymmetric spectrum observed in the heliosheath; the solar wind in the outer heliospheremay exhibit strong asymmetric scaling, where the spectrum is prevalently right-skewed. Theobtained delay between Voyager 1 and 2 can certainly be correlated with the evolution of theheliosphere, providing an additional support to some earlier independent claims that the solarwind termination shock itself is possibly asymmetric (Stone et al., 2008).

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• W. M. Macek, Multifractality and Intermittency in the Solar Wind, Nonlin. Processes Geophys. 14, 695–700(2007).

• W. M. Macek and A. Szczepaniak, Generalized Two-Scale Weighted Cantor Set Model for Solar WindTurbulence, Geophys. Res. Lett., 35, L02108 (2008).

• A. Szczepaniak and W. M. Macek, Asymmetric Multifractal Model for Solar Wind Intermittent Turbulence,Nonlin. Processes Geophys., 15, 615–620 (2008).

• W. M. Macek, Chaos and Multifractals in the Solar System Plasma, in Topics on Chaotic Systems, edited by C.H. Skiadas et al., Selected Papers from Chaos 2008 International Conference, World Scientific, New Jersey,pp. 214–223 (2009).

• W. M. Macek and A. Wawrzaszek, Evolution of Asymmetric Multifractal Scaling of Solar Wind Turbulence inthe Outer Heliosphere, J. Geophys. Res., 114, A03108 (2009).

• W. M. Macek, A. Wawrzaszek, T. Hada, Multiscale Multifractal Intermittent Turbulence in Space Plasmas, J.Plasma Fusion Res. Series, vol. 8, 142-147 (2009).

• W. M. Macek, Chaos and Multifractals in the Solar Wind, Adv. Space Res., 46, 526–531 (2010).• W. M. Macek and A. Wawrzaszek, Multifractal Turbulence at the Termination Shock, Solar Wind 12, American

Institute of Physics, vol. CP1216 (2010), pp. 572–575.• H. Lamy, A. Wawrzaszek, W. M. Macek, T. Chang, New Multifractal Analyses of the Solar Wind Turbulence:

Rank-Ordered Multifractal Analysis and Generalized Two-scale Weighted Cantor Set Model, Solar Wind 12,American Institute of Physics, vol. CP1216 (2010), pp. 124–127.

• A. Wawrzaszek and W. M. Macek, Observation of Multifractal Spectrum in Solar Wind Turbulence by Ulyssesat High Latitudes, J. Geophys. Res., 115, A07104 (2010).

• W. M. Macek and A. Wawrzaszek, Multifractal Structure of Small and Large Scales Fluctuations ofInterplanetary Magnetic Fields, Planet. Space Sci., 59, 569–574 (2011a).

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• W. M. Macek and A. Wawrzaszek, Multifractal Two-Scale Cantor Set Model for Slow Solar Wind Turbulence inthe Outer Heliosphere During Solar Maximum, Nonlin. Processes Geophys., 18, 287-294 (2011b).

• W. M. Macek, A. Wawrzaszek, V. Carbone, Observation of the Multifractal Spectrum at the Termination Shockby Voyager 1, Geophys. Res. Lett., 38, L19103 (2011).

• W.M. Macek, A. Wawrzaszek, V. Carbone, Observation of the Multifractal Spectrum in the Heliosphere andthe Heliosheath by Voyager 1 and 2, Journal of Geophysical Research, 117, A12101 (2012).

• W. M. Macek, Multifractal Turbulence in the Heliosphere, in Exploring the Solar Wind, edited by M. Lazar,Intech, ISBN 978-953-51-0399-4 (2012).

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Fractal

A measure (volume) V of a set as afunction of size l

V (l)∼ lDF

The number of elements of size l neededto cover the set

N(l)∼ l−DF

The fractal dimension

DF = liml→0

lnN(l)ln1/l

Multifractal

A (probability) measure versus singularitystrength, α

pi(l) ∝ lαi

The number of elements in a small rangefrom α to α + dα

Nl(α)∼ l− f (α)

The multifractal singularity spectrum

f (α) = limε→0

liml→0

ln[Nl(α+ ε)−Nl(α− ε)]

ln1/l

The generalized dimension

Dq =1

q−1liml→0

ln∑Nk=1(pk)

q

ln l

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Multifractal Characteristics

Fig. 1. (a) The generalized dimensions Dq as a function of any real q, −∞ < q < ∞,and (b) the singularity multifractal spectrum f (α) versus the singularity strength α withsome general properties: (1) the maximum value of f (α) is D0; (2) f (D1) = D1; and (3)the line joining the origin to the point on the f (α) curve where α = D1 is tangent to thecurve (Ott et al., 1994).

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Generalized Scaling Property

The generalized dimensions are important characteristics of complex dynamical systems; theyquantify multifractality of a given system (Ott, 1993). In the case of turbulence cascadethese generalized measures are related to inhomogeneity with which the energy is distributedbetween different eddies (Meneveau and Sreenivasan, 1991). In this way they provideinformation about dynamics of multiplicative process of cascading eddies. Here high positivevalues of q> 1 emphasize regions of intense fluctuations larger than the average, while negativevalues of q accentuate fluctuations lower than the average (cf. Burlaga 1995).

Using (∑ piq ≡ 〈pi

q−1〉av) a generalized average probability measure

µ(q, l)≡ q−1√〈(pi)q−1〉av (1)

we can identify Dq as scaling of the measure with size l

µ(q, l) ∝ lDq (2)

Hence, the slopes of the logarithm of µ(q, l) of Eq. (2) versus log l (normalized) provides

Dq = liml→0

log µ(q, l)log l

(3)

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Measures and Multifractality

Similarly, we define a one-parameter q family of (normalized) generalized pseudoprobabilitymeasures (Chhabra and Jensen, 1989; Chhabra et al., 1989)

µi(q, l)≡pq

i (l)

∑Ni=1 pq

i (l)(4)

Now, with an associated fractal dimension index fi(q, l) ≡ logµi(q, l)/ log l for a given q themultifractal singularity spectrum of dimensions is defined directly as the average taken withrespect to the measure µ(q, l) in Eq. (4) denoted by 〈. . .〉

f (q) ≡ liml→0

N

∑i=1

µi(q, l) fi(q, l) = liml→0

〈logµi(q, l)〉log(l)

(5)

and the corresponding average value of the singularity strength is given by(Chhabra and Jensen, 1987)

α(q) ≡ liml→0

N

∑i=1

µi(q, l) αi(l) = liml→0

〈log pi(l)〉log(l)

. (6)

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Methods of Data Analysis

Magnetic Field Strength Fluctuations and Generalized Measures

Given the normalized time series B(ti), where i = 1, . . . ,N = 2n (we take n = 8), to each intervalof temporal scale ∆t (using ∆t = 2k, with k = 0,1, . . . ,n) we associate some probability measure

p(x j, l)≡1N

j∆t

∑i=1+( j−1)∆t

B(ti) = p j(l), (7)

where j = 2n−k, i.e., calculated by using the successive (daily) average values 〈B(ti,∆t)〉 of B(ti)between ti and ti +∆t. At a position x = vswt, at time t, where vsw is the average solar windspeed, this quantity can be interpreted as a probability that the magnetic flux is transferred to asegment of a spatial scale l = vsw∆t (Taylor’s hypothesis).

The average value of the qth moment of the magnetic field strength B should scale as

〈Bq(l)〉 ∼ lγ(q), (8)

with the exponent γ(q) = (q−1)(Dq−1) as shown by Burlaga et al. (1995).

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Mutifractal Models for Turbulence

Fig. 1. Generalized two-scale Cantor set model for turbulence (Macek, 2007).

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Solutions

Transcendental equation (for n→ ∞)

pq

lτ(q)1

+(1− p)q

lτ(q)2

= 1 (9)

For l1 = l2 = λ and any q in Eq. (9) one has for the generalized dimensions

τ(q)≡ (q−1)Dq =ln[pq+(1− p)q]

ln λ. (10)

Space filling turbulence (λ = 1/2):the multifractal cascade p−model for fully developed turbulence,the generalized weighted Cantor set (Meneveau and Sreenivasan, 1987).The usual middle one-third Cantor set (without any multifractality):p = 1/2 and λ = 1/3.

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Degree of Multifractality and Asymmetry

The difference of the maximum and minimum dimension (the least dense and most densepoints in the phase space) is given, e.g., by Macek (2006, 2007)

∆≡ αmax−αmin = D−∞−D∞ =

∣∣∣∣log(1− p)log l2

− log(p)log l1

∣∣∣∣. (11)

In the limit p→ 0 this difference rises to infinity (degree of multifractality).

The degree of multifractality ∆ is simply related to the deviation from a simple self-similarity.That is why ∆ is also a measure of intermittency, which is in contrast to self-similarity (Frisch,1995, chapter 8).

Using the value of the strength of singularity α0 at which the singularity spectrum has itsmaximum f (α0) = 1 we define a measure of asymmetry by

A≡ α0−αmin

αmax−α0. (12)

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Schematic of the Heliospheric Boundaries.

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Fig. 2. The heliospheric distances from the Sun and the heliographic latitudes during eachyear of the Voyager mission. Voyager 1 and 2 spacecraft are located above and below the solarequatorial plane, respectively.

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Voyager SpacecraftV1

7 – 60 AU (1980 – 1995)70 – 90 AU (1999 – 2003)95 – 107 AU (2005 – 2008)

V2

7 – 40 AU (1980 – 1990)60 – 80 AU (2002 – 2006)85 – 90 AU (2008 – 2009)

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Fig. 3. The singularity spectrum f (α) as a function of a singularity strength α. The values are calculated for theweighted two-scale (continuous lines) model and the usual one-scale (dashed lines) p-model with the parametersfitted using the magnetic fields (diamonds) measured by Voyager 2 in the near heliosphere at (a) 6–8 AU (b) 16–18AU, (c) 31–33 AU, and (d) 45–47 AU, correspondingly (Macek et al. 2012).

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Fig. 4. The singularity spectrum f (α) as a function of a singularity strength α. The values are calculatedfor the weighted two-scale (continuous lines) model and the usual one-scale (dashed lines) p-model with theparameters fitted using the magnetic fields (diamonds) measured by Voyager 2 in the distant heliosphere at variousdistances before crossing the termination shock, (a) 57–59 AU (b) 69–71 AU, (c) 75–77 AU, and (d) 78–81 AU,correspondingly (Macek et al. 2012).

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Fig. 5. The singularity spectrum f (α) as a function of a singularity strength α. The values are calculatedfor the weighted two-scale (continuous lines) model and the usual one-scale (dashed lines) p-model with theparameters fitted using the magnetic fields (diamonds) measured by Voyager 1 in the heliosheath at variousheliocentric distances of (a) 94–97 AU and (c) 105–107 AU AU, and Voyager 2 at (b) 85–88 AU and (d) 88–90 AU,correspondingly (Macek et al. 2012).

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Fig. 6. The degree of multifractality ∆ in the heliosphere as a function of the distances from the Sun fitted to a periodically decreasing

function shown by a continuous line during solar minimum (MIN) and solar maximum (MAX), declining (DEC) and rising (RIS) phases of

solar cycles. The crossing of the termination shock by Voyager 2 is marked by a vertical dashed line (TS). Below are shown the Sunspot

Numbers (SSN) during the period of 1980–2009 (Macek et al. 2012).

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Fig. 7. The degree of asymmetry A of the multifractal spectrum in the heliosphere as a function of the heliosphericdistances during solar minimum (MIN) and solar maximum (MAX), declining (DEC) and rising (RIS) phases of solarcycles. The value A = 1 corresponds to the one-scale symmetric model (dotted). The crossing of the terminationshock by Voyager 2 is marked by a vertical (dashed) line, TS (Macek et al. 2012).

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Fig. 8. The degree of multifractality ∆ in the heliosphere versus the heliospheric distances for Voyager 1 and 2with fits to straight lines. The crossings of the termination shock (TS) by both spacecraft are marked by verticaldashed lines. (Macek et al. 2012).

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Fig. 9. The degree of asymmetry A of the multifractal spectrum in the heliosphere as a function of the heliosphericdistance; the value A= 1 (dotted) corresponds to the one-scale symmetric model. The crossings of the terminationshock (TS) by Voyager 1 and 2 are marked by vertical dashed lines (Macek et al. 2012).

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Fig. 10. Comparison of the periodic parts of functions shown in case (a), (corresponding to Figure 6) fitted tothe obtained degree of multifractality as functions of heliocentric distances (b) and time (c) for Voyager 1 (dashedlines) and 2 (continuous lines) (Macek et al. 2012).

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Figure 1: Degree of multifractality ∆ and asymmetry A for the magnetic fieldstrengths in the outer heliosphere and beyond the termination shock (Macek etal. 2011).

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Conclusions

• Using our weighted two-scale Cantor set model, which is a convenient tool to investigate theasymmetry of the multifractal spectrum, we confirm the characteristic shape of the universalmultifractal singularity spectrum. f (α) is a downward concave function of scaling indices α.

• For the first time we show that the degree of multifractality for magnetic field fluctuationsof the solar wind falls steadily with the distance from the Sun and seems to be modulatedby the solar activity with a time delay of several years, corresponding to a difference ofdistances of about 10 AU in the very distant heliosphere.

• Moreover, we have again observed a change of the asymmetry of the spectrum whencrossing the termination shock by Voyager 1 and 2. Consequently, a concentration ofmagnetic fields shrinks (stretches) resulting in thinner (fatter) flux tubes or stronger (weaker)current concentration in the heliosheath. Admittedly, we can still have approximatelysymmetric spectrum in the heliosheath, where the plasma is expected to be roughly inequilibrium in the transition to the interstellar medium.

• We also confirm our earlier results that before the shock crossing, especially during solarmaximum, turbulence is more multifractal than that in the heliosheath (Macek et al., 2011).

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• In addition, in the whole heliosphere during solar minimum the spectrum is dominated byvalues of α > 1, which could change when crossing the termination heliospheric shock. Thismay represents a first direct in situ information of interest in the astrophysical context. Infact, a density of measure dominated by α < 1 (α > 1) would imply that the magnetic fieldin the very local interstellar medium is roughly confined in thin (thick) filaments of high (low)magnetic density.

• That means that it would be interesting to investigate whether the heliosheath is dominatedby ’voids’ of magnetic fields in the very local interstellar medium related to the presence ofvery localized intense magnetic field structures (cf. Frisch, 1995).

• Such informations, obtained in situ and not only remotely (through scintillationsobservations), are important in the context of question of interstellar turbulence and stellarformation modeling (e.g., Spangler et al. 2009). Finally, it is worth mentioning that ouranalysis brings significant additional support to earlier results suggesting that the solar windtermination shock is asymmetric (Stone et al., 2008).

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Epilogue

Within the complex dynamics of the solarwind’s and interstellar medium fluctuatingintermittent plasma parameters, there isa detectable, hidden order described bya Cantor set that exhibits a multifractalstructure.

This means that the observedintermittent behavior of magneticfluctuations in the heliosphere and thevery local interstellar medium may resultfrom intrinsic nonlinear dynamics ratherthan from random external forces.

Thank you!

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This work was supported by the European Community’s Seventh FrameworkProgramme ([FP7/2007–2013]) under Grant agreement No. 313038/STORM andcosponsored by the Ministry of Science and Higher Education (2013–2015), Poland(MNiSW).

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